Question

Find the points at which the function \(f\) given by \(f(x)=(x-2)^4(x+1)^3\) has (i)local maxima (ii)local minima (iii)point of inflexion

Answer 1

The given function is \(f(x)=(x-2)^4(x+1)^3\)
\(∴f'(x)=4(x-2)^3(x+1)^3+3(x+1)^2(x-2)^4\)
\(=(x-2)^3(x+1)^2[4(x+1)+3(x-2)]\)
\(=(x-2)^3(x+1)^2(7x-2)\)
Now,\(f'(x)=0\)
\(⇒x=-1\) and \(x=\frac{2}{7}\, or\, x=2\)
Now, for values of \(x\) close to \(\frac{2}{7}\) and to the left of \(\frac{2}{7}\, f'(x)>0.\) Also, for values of \(x\) close to \(\frac{2}{7}\) and to the left of \(\frac{2}{7}\, f'(x)<0.\)
Thus \(x=\frac{2}{7}\) is the point of local maxima.
Now, for values of \(x\) close to \(2\) and to the left of \(2,f'(x)<0.\)
Also, for values of \(x\) close to \(2\) and to the right of \(2, f'(x)>0\).
Thus, \(x = 2\) is the point of local minima.
Now, as the value of \(x\) varies through \(−1,F'(X)\) does not changes its sign. Thus, \(x=−1\) is the point of inflexion.